By using Theorem \(3.2\)  of [1]}, the Taylor's expansion
of \(\beta _{k}^{\prime }s\)  are
\(\beta _{0} & = & \xi \mid \Delta xf_{j+\frac{1}{2}}^{(1)}-\frac{23}{24}\Delta x^{3}f_{j+\frac{1}{2}}^{(3)}\mid +\mid \Delta x^{2}f_{j+\frac{1}{2}}^{(2)}-\frac{3}{2}\Delta x^{3}f_{j+\frac{1}{2}}^{(3)}\mid +O(\Delta x^{4}),\nonumber \\\beta _{1} & = & \xi \mid \Delta xf_{j+\frac{1}{2}}^{(1)}+\frac{1}{24}\Delta x^{3}f_{j+\frac{1}{2}}^{(3)}\mid +\mid \Delta x^{2}f_{j+\frac{1}{2}}^{(2)}-\frac{1}{2}\Delta x^{3}f_{j+\frac{1}{2}}^{(3)}\mid +O(\Delta x^{4}),\\\beta _{2} & = & \xi \mid \Delta xf_{j+\frac{1}{2}}^{(1)}+\frac{1}{24}\Delta x^{3}f_{j+\frac{1}{2}}^{(3)}\mid +\mid \Delta x^{2}f_{j+\frac{1}{2}}^{(2)}+\frac{1}{2}\Delta x^{3}f_{j+\frac{1}{2}}^{(3)}\mid +O(\Delta x^{4}).\nonumber \) 
